On theories for quasi-inductive definitions

On theories for quasi–inductive definitions

Riccardo Bruni

Department of PhilosophyUniversity of Florence

[email protected]

Leeds Proof theory SeminarApril 28/2010

Taming circularity

The semantics by revision

Assume we have a (for simplicity, unary) predicate G defined by:

G (x) =Def ϕ(x ,G )

This definitions is circular in a straightforward sense, henceillegitimate (i.e., it is not possible to give an extension to thepredicate G ).

Do it in a revision–style:

1. give G a tentative extension, say h := a1, . . . , an, . . .2. revise h by ϕ(x ,G ) itself, namely

h 7→ h′ := a | ϕ(a,G := h)

3. fix a rule to single out regularities (i.e. recurringelements/hypotheses), obtained by iterating 1-2.

1. give G a tentative extension, say h := a1, . . . , an, . . .

2. revise h by ϕ(x ,G ) itself, namely

h 7→ h′ := a | ϕ(a,G := h)

To illustrate:

Let ϕ(x ,G ) be

[x = a ∨ (x = b ∧ ¬G (x))]

The revision step then gives

Input Output∅ ⇒ a, ba, b ⇒ aa ⇒ a, bb ⇒ a

......

To illustrate:

Let ϕ(x ,G ) be

[x = a ∨ (x = b ∧ ¬G (x))]

......

To illustrate:

Let ϕ(x ,G ) be

[x = a ∨ (x = b ∧ ¬G (x))]

......

To illustrate:

Let ϕ(x ,G ) be

[x = a ∨ (x = b ∧ ¬G (x)/x 6∈ X )]

......

To illustrate:

Let ϕ(x ,G ) be

[x = a ∨ (x = b ∧ ¬G (x))]

......

To illustrate:

Let ϕ(x ,G ) be

[x = a ∨ (x = b ∧ ¬G (x))]

......

Alternative views

Herzberger’s(the ‘pure’ theory)

arithmetical

liminf

Gupta’s(the ‘corrected’ th.)

Z ⊆ N

liminf ∪ h0

Belnap’s(the ‘maximal’ th.)

Z ⊆ N

X |X coherent†

†X ⊆ N s. t., for λ limit

h+<λ ⊆ X X ∩ h−<λ = ∅

with h+/−<λ sets of positive/negative stable elements (below λ).

Alternative views

Herzberger’s(the ‘pure’ theory)

arithmetical

liminf

Gupta’s(the ‘corrected’ th.)

Z ⊆ N

liminf ∪ h0

Belnap’s(the ‘maximal’ th.)

Z ⊆ N

X |X coherent†

†X ⊆ N s. t., for λ limit

h+<λ ⊆ X X ∩ h−<λ = ∅

with h+/−<λ sets of positive/negative stable elements (below λ).

On quasi–induction

The quasi–inductive schema

Let Γ : P(N)→ P(N) operator whatsoever.

A quasi–inductive sequence 〈Γα | α ∈ ON〉 of sets Γα ⊆ N, is theone given by the clauses

Γ0 = ∅Γα+1 = Γ(Γα)

Γλ = lim infβ→λ Γβ, λ limit

with lim infβ→λ Γβ := n | ∃α < λ∀β < λ(α ≤ β → n ∈ Γβ)

The quasi–inductive schema

Let Γ : P(N)→ P(N) operator whatsoever.

A quasi–inductive sequence 〈Γα | α ∈ ON〉 of sets Γα ⊆ N, is theone given by the clauses

Γ0 = ∅Γα+1 = Γ(Γα)

Γλ = lim infβ→λ Γβ, λ limit

with lim infβ→λ Γβ := n | ∃α < λ∀β < λ(α ≤ β → n ∈ Γβ)

Basic properties

Γ0 = ∅Γα+1 = Γ(Γα)

Γλ = lim infβ→λ Γβ

I for (Γ+∞, Γ

−∞) stability pair of the sequence of sets, with

Γ+∞ := lim inf

α→∞Γα = n | ∃α∀β(α ≤ β → n ∈ Γβ)

Γ−∞ := lim infα→∞

(N \ Γα) = n | ∃α∀β(α ≤ β → n 6∈ Γβ)

one shows that there are (limit) stabilization ordinals δs, forwhich Γδ = Γ+

∞ and Γ−δ = lim infα→δ(N \ Γα) = Γ−∞;

I further, one shows that levels as such appear again and againin the quasi–inductive iteration of the operator, according to agiven period. That is, for a stabilization ordinal δ one provesthat

∃β∀γ[Γδ+βγ = Γδ]

Basic properties

Γ0 = ∅Γα+1 = Γ(Γα)

I for (Γ+∞, Γ

Γ+∞ := lim inf

α→∞Γα = n | ∃α∀β(α ≤ β → n ∈ Γβ)

(N \ Γα) = n | ∃α∀β(α ≤ β → n 6∈ Γβ)

∞ and Γ−δ = lim infα→δ(N \ Γα) = Γ−∞;I further, one shows that levels as such appear again and again

in the quasi–inductive iteration of the operator, according to agiven period. That is, for a stabilization ordinal δ one provesthat

Basic properties

Γ0 = ∅Γα+1 = Γ(Γα)

I for (Γ+∞, Γ

Γ+∞ := lim inf

α→∞Γα = n | ∃α∀β(α ≤ β → n ∈ Γβ)

(N \ Γα) = n | ∃α∀β(α ≤ β → n 6∈ Γβ)

∞ and Γ−δ = lim infα→δ(N \ Γα) = Γ−∞;

I further, one shows that levels as such appear again and againin the quasi–inductive iteration of the operator, according to agiven period. That is, for a stabilization ordinal δ one provesthat

Basic properties

Γ0 = ∅Γα+1 = Γ(Γα)

I for (Γ+∞, Γ

Γ+∞ := lim inf

α→∞Γα = n | ∃α∀β(α ≤ β → n ∈ Γβ)

(N \ Γα) = n | ∃α∀β(α ≤ β → n 6∈ Γβ)

∞ and Γ−δ = lim infα→δ(N \ Γα) = Γ−∞;I further, one shows that levels as such appear again and again

in the quasi–inductive iteration of the operator, according to agiven period. That is, for a stabilization ordinal δ one provesthat

Axioms

A family of theories for QIDs

Given LAr , the language of Peano Arithmetic, L0 is the oneextending its alphabet by

I a second sort of individual variables α, β, . . . for ordinalnumbers;

I individual constants 0Ω, ω;

I symbols for functions succΩ,+Ω,×Ω on the ordinals;

I a predicate constant <Ω for the ordering.

Language: operator forms

I for L = LAr ∪ X 1 (X fresh unary predicate variable), anoperator form is a formula A(x ,X ) with displayedfree–variables;

I we obtain our language L(K) by adding to L0 constantpredicate symbols HA for every operator form A(x ,X ) in L,with logical complexity K (K = ∆n,Πn,Σn, n ∈ N);

I the notion of ‘formula’ for the expanded language must bere–defined so to comprise atoms HA(n, α) (abbrev. n ∈ HαA)with n ∈ TERMN and α ∈ TERMΩ.

Language: conventions

(HαA ≡ HβA) :≡ ∀x(x ∈ HαA ↔ x ∈ HβA)

x ∈ H+A (∞) :≡ ∃β∀δ(β ≤ δ → x ∈ HδA)

x ∈ H−A (∞) :≡ ∃β∀δ(β ≤ δ → x 6∈ HδA)

x ∈ H+A (λ) :≡ ∃β < λ∀δ < λ(β ≤ δ → x ∈ HδA)

x ∈ H−A (λ) :≡ ∃β < λ∀δ < λ(β ≤ δ → x 6∈ HδA)

Axioms

For a fixed K = ∆n,Πn,Σn, the axioms of QID(K) amount at:

I a complete axiomatization of first–order classical logic withequality

I the axioms of arithmetic (with CI)

Axioms: logical

Axioms: logical, arithmetical

Axioms: logical, arithmetical, ordinal–theoretical

I standard assumptions on the ordering <Ω, on ordinalindividual constants, the defining equations of the stock ofprimitive ordinal functions as well as axioms on their basicproperties (monotonicity, inverses), plus a schema oftransfinite induction

Axioms: QID

(QID.1) x ∈ H0A → x 6= x

(QID.2) x ∈ Hα+1A ↔ A(x ,Hα

(QID.3) Lim(λ)→ [x ∈ HλA ↔ (∃α < λ)(∀β < λ)(α ≤ β → x ∈ Hβ

(QID.4) ∀α∃λ(Lim(λ)∧α < λ∧(H+A (λ) ≡ H+

A (∞))∧(H−A (λ) ≡ H−A (∞)))

Axioms: QID

(QID.1) x ∈ H0A → x 6= x

(QID.2) x ∈ Hα+1A ↔ A(x ,Hα

(QID.3) Lim(λ)→ [x ∈ HλA ↔ (∃α < λ)(∀β < λ)(α ≤ β → x ∈ Hβ

(QID.4) ∀α∃λ(Lim(λ)∧α < λ∧(H+A (λ) ≡ H+

A (∞))∧(H−A (λ) ≡ H−A (∞)))

Some results

Periodicity

PROPOSITION

Let σ be any limit ordinal such that HσA ≡ H+∞A and H−σA ≡ H−∞A ,

for a K–operator form A(x ,X ). Then QID(K) proves that thereexists a unique ordinal p(σ) > 0, the period of σ, such that:

(i) for every ordinal γ, HσA ≡ Hσ+p(σ)γA

(ii) for every ordinal α > σ there exists an ordinal0 ≤ ν < p(σ) such that HαA ≡ H

σ+νA

Periodicity

PROPOSITION

Let σ be any limit ordinal such that HσA ≡ H+∞A and H−σA ≡ H−∞A ,

for a K–operator form A(x ,X ). Then QID(K) proves that thereexists a unique ordinal p(σ) > 0, the period of σ, such that:

(i) for every ordinal γ, HσA ≡ Hσ+p(σ)γA

(ii) for every ordinal α > σ there exists an ordinal0 ≤ ν < p(σ) such that HαA ≡ H

σ+νA

Lower bound: Are QIDs a next natural step?

Theories for first-order nonmonotone inductive definitions FID(K)are defined in a similar manner as our QID(K) (with relationsPαA(n) for an operator form A(x ,X ) whatsoever), except that theoperator axioms are

FID(K)

(OP.1) PαA(s)↔ P<αA (s) ∨ A(s,P<αA )(OP.2) A(s,P∞A )→ P∞A (s)

[where P<αA (s) := (∃β < α)PβA(s), and P∞A (s) := ∃βPβA(s)]

Theories for first-order nonmonotone inductive definitions FID(K)are defined in a similar manner as our QID(K) (with relationsPαA(n) for an operator form A(x ,X ) whatsoever), except that theoperator axioms are

FID(K)

(OP.1) PαA(s)↔ P<αA (s) ∨ A(s,P<αA )(OP.2) A(s,P∞A )→ P∞A (s)

[where P<αA (s) := (∃β < α)PβA(s), and P∞A (s) := ∃βPβA(s)]

PROPOSITION

For every K, there exists a formula–to–formula translation (·)′ fromLK

FID to L(K) such that, for every formula A of LKFID we have:

`FID(K) A⇒ `QID(K) A′

(Set–theoretic) Upper bound

Let T be KP+(∆2–SEP)+(∆3–COLL). Then,

PROPOSITION

The theory for arithmetical QIDs, QID(Π∞), is embeddable in T.

Let T be KP+(∆2–SEP)+(∆3–COLL). Then,

PROPOSITION

The theory for arithmetical QIDs, QID(Π∞), is embeddable in T.

FIRST: use Σ–recursion.

QHA(α, f ) :=

Fun(f ) ∧ dom(f ) = α∧∧(∀β < α)[(β = 0 ∧ f (β) = 0)∨∨(∃γ < α)(β = γ + 1∧f (β)=z ∈ N |AN(z , f (γ)))∨∨(Lim(β) ∧ f (β) =

⋃γ<β

⋂γ≤δ<β f (δ))]

x ∈ HαA

:= ∃f [QHA(α + 1, f ) ∧ x ∈ f (α)]

x ∈ H−αA := (∃β < α)(∀γ < α)(β ≤ γ → x 6∈ HγA)

x ∈ H+∞A := ∃β∀γ(β ≤ γ → x ∈ Hγ

x ∈ H−∞A := ∃β∀γ(β ≤ γ → x 6∈ HγA)

QHA(α, f ) :=

⋃γ<β

⋂γ≤δ<β f (δ))]

x ∈ HαA

:= ∃f [QHA(α + 1, f ) ∧ x ∈ f (α)]

x ∈ H−αA := (∃β < α)(∀γ < α)(β ≤ γ → x 6∈ HγA)

x ∈ H+∞A := ∃β∀γ(β ≤ γ → x ∈ Hγ

x ∈ H−∞A := ∃β∀γ(β ≤ γ → x 6∈ HγA)

QHA(α, f ) :=

⋃γ<β

⋂γ≤δ<β f (δ))]

x ∈ HαA := ∃f [QHA(α + 1, f ) ∧ x ∈ f (α)]

x ∈ H−αA := (∃β < α)(∀γ < α)(β ≤ γ → x 6∈ HγA)

x ∈ H+∞A := ∃β∀γ(β ≤ γ → x ∈ Hγ

x ∈ H−∞A := ∃β∀γ(β ≤ γ → x 6∈ HγA)

QHA(α, f ) :=

⋃γ<β

⋂γ≤δ<β f (δ))]

x ∈ HαA := ∃f [QHA(α + 1, f ) ∧ x ∈ f (α)]

x ∈ H−αA := (∃β < α)(∀γ < α)(β ≤ γ → x 6∈ HγA)

x ∈ H+∞A := ∃β∀γ(β ≤ γ → x ∈ Hγ

x ∈ H−∞A := ∃β∀γ(β ≤ γ → x 6∈ HγA)

For all operator forms A(x ,X ), KP proves:

1. ∀α∃f QHA(α, f ).

2. QHA(α, f ) ∧ β < α→ QHA(β, f )

3. QHA(α, f ) ∧ QHA(β, g) ∧ α ≤ β → (∀γ < α)(f (γ) = g(γ))

4. n ∈ N→ (n ∈ Hα+1A ↔ AN(n,Hα

5. n ∈ N ∧ Lim(λ)→ (n ∈ HλA ↔ (∃α < λ)(∀β < λ)(α ≤ β →

n ∈ HβA))

Stages HαAs can be equivalently described by means of the Π1

condition

x ∈ HαA ↔ ∀f [QHA(f , α + 1)→ x ∈ f (α)]

Hence, formulas x ∈ HαA are ∆T

1 , while x ∈ H+∞A , x ∈ H−∞A are

both ΣT2 .

For all operator forms A(x ,X ), KP proves:

1. ∀α∃f QHA(α, f ).

2. QHA(α, f ) ∧ β < α→ QHA(β, f )

3. QHA(α, f ) ∧ QHA(β, g) ∧ α ≤ β → (∀γ < α)(f (γ) = g(γ))

4. n ∈ N→ (n ∈ Hα+1A ↔ AN(n,Hα

5. n ∈ N ∧ Lim(λ)→ (n ∈ HλA ↔ (∃α < λ)(∀β < λ)(α ≤ β →

n ∈ HβA))

Stages HαAs can be equivalently described by means of the Π1

condition

x ∈ HαA ↔ ∀f [QHA(f , α + 1)→ x ∈ f (α)]

Hence, formulas x ∈ HαA are ∆T

1 , while x ∈ H+∞A , x ∈ H−∞A are

both ΣT2 .

SECOND: use Π2–collection.

PROPOSITION (Covering)

In T it is provable that, for every ordinal α, there exists a limitordinal δ > α such that H+∞

A ⊆ HδA, H−∞A ⊆ H−δA , Hδ

A ∩ H−∞A = ∅and H−δA ∩ H+∞

A = ∅.

THIRD: use Σ2–collection.

PROPOSITION (Stability)

In T it is provable that, for every arithmetical operator formA(x ,X ), ∀α∃λ(α < λ ∧ Hλ

A ≡ H+∞A ∧ H−λA ≡ H−∞A ).

THIRD: use Σ2–collection.

PROPOSITION (Stability)

In T it is provable that, for every arithmetical operator formA(x ,X ), ∀α∃λ(α < λ ∧ Hλ

A ≡ H+∞A ∧ H−λA ≡ H−∞A ).

Final Remarks

Future work

I Upper bound refinement (P. Welch: use Σ2–KP)

I Is this an exact bound?

Future work

I Upper bound refinement (P. Welch: use Σ2–KP)

I Is this an exact bound?

The end

The limit rule and why we need it

Let instead G be T , and our definition be

T (x) := (x = ps = tq ∧ val(s) = val(t)) ∨∨ (x = pT (s)q ∧ T (pT (s)q)) ∨∨ (x = p¬ϕq ∧ ¬T (pϕq)) ∨

ϕ(x ,T )

Namely: let ϕ(x ,T ) be the arithmetical formula defining aTarskian truth predicate.

Furthermore, let F be the set:

F := ϕ0, ϕ1, . . . , ϕn, . . .

ϕ0 := >ϕn+1 := Tϕn

Assume that the revision process is made out of successive step,everyone using the previous output as input, starting from h0 = ∅.

Then, the calculation of the extension of the truth predicate yields:

Input Output∅ ⇒ ϕ0ϕ0 ⇒ ϕ0, ϕ1ϕ0, ϕ1 ⇒ ϕ0, ϕ1, ϕ2ϕ0, ϕ1, ϕ2 ⇒ ϕ0, ϕ1, ϕ2, ϕ3

......

PROBLEM: cannot conclude F ⊆ T by finitary means.

Assume that the revision process is made out of successive step,everyone using the previous output as input, starting from h0 = ∅.Then, the calculation of the extension of the truth predicate yields:

......

Assume that the revision process is made out of successive step,everyone using the previous output as input, starting from h0 = ∅.Then, the calculation of the extension of the truth predicate yields:

......

Ordinal axioms

(Ω.1) ∀αβ(α = β ∨ α < β ∨ β ∨ α)

(Ω.2) ∀α(¬α < α)

(Ω.3) ∀αβγ(α < β ∧ β < γ → α < γ)

(Ω.4) ∀α(0Ω ≤ α) [where α ≤ β := (α < β ∨ α = β)]

(Ω.5) ∀α(α < α′) [with α′ = succΩ(α)]

(Ω.6) ∀αβ(α < β → α′ ≤ β)

(Ω.7) 0Ω < ω ∧ ∀α < ω(α′ < ω)

(Ω.8) ∀λ(Lim(λ)→ ω ≤ λ)

[where Lim(α) := (0 < α ∧ ∀β < α(β′ < α))]

Ordinal axioms

(Ω.9) ∀α(α + 0Ω = α)

(Ω.10) ∀αβ(α + β′ = (α + β)′)

(Ω.11) ∀αβγ(α < β → γ + α < γ + β)

(Ω.12) ∀αβγ(α ≤ β → α + γ ≤ β + γ)

(Ω.13) ∀α(α0Ω = 0Ωα = 0Ω)

(Ω.14) ∀αβ(αβ′ = αβ + α)

(Ω.15) ∀αβγ(0Ω < γ ∧ α < β → γα < γβ)

(Ω.16) ∀αβγ(α ≤ β → αγ ≤ βγ)

(Ω.17) ∀αβ(α < β → ∃γ ≤ β(α + γ = β))

(Ω.18) ∀αβ(0Ω < β → ∃γ ≤ α∃δ < β(α = βγ + δ))

Ordinal axioms

(L(K)− IN) A(0) ∧ ∀x(A(x)→ A(x ′))→ ∀xA(x)

(L(K)− IΩ) ∀α((∀β < α)A(β)→ A(α))→ ∀αA(α)

Lower bound strategy

1. A(x ,X )w.l.g7−→ B(x ,X ) := (X (x) ∨ A(x ,X ));

2. Verify axioms (OP.1-2) for levels Hα/+∞B and inflationary

B(x ,X )s (using inclusivity: C (s)→ B(s,C ), and stability);

3. Define the embedding in the expected manner withPαA 7−→ HαB .

1. A(x ,X )w.l.g7−→ B(x ,X ) := (X (x) ∨ A(x ,X ));

Proof of the Covering Lemma

Since(∀x ∈ N)∃β(x ∈ H+∞

A → (∀γ ≥ β)(x ∈ HγA))

is a simple consequence of the definitions, (COLL) ensures thenthat

∃b(∀x ∈ N)(∃β ∈ b)(x ∈ H+∞A → (∀γ ≥ β)(x ∈ Hγ

A)) (1)

By (SEP), put b′ = β ∈ b | (∃n ∈ N)(∀γ ≥ β)(n ∈ HγA).

Find sets c, c ′ playing for H−∞A the role b and b′ play for H+∞A .

Finally, let α be any ordinal. Take δ to be the least limit ordinalsuch that ξ < δ where ξ = α ∪ b′ ∪ c ′. It’s easy to see that δsatisfies the lemma. Q.E.D.

A → (∀γ ≥ β)(x ∈ HγA))

∃b(∀x ∈ N)(∃β ∈ b)(x ∈ H+∞A → (∀γ ≥ β)(x ∈ Hγ

A)) (1)

A → (∀γ ≥ β)(x ∈ HγA))

∃b(∀x ∈ N)(∃β ∈ b)(x ∈ H+∞A → (∀γ ≥ β)(x ∈ Hγ

A)) (1)

Proof of the Stability Lemma: Overview

1. Ordinals δs are ‘almost good’, except that they may contain

elements outside H+/−∞A ;

2. The set W = x ∈ N | Uδ(x) of them (w.r.t. to a fixed δ),admits a Σ1–definition;

3. This is used for defining a ∆2 recursive enumeration F of W ,with every x ∈W occurring infinitely often

4. By (Σ2–)recursion again, one defines a sequence

〈HβA | β < µ〉, with µ limit, H0

A = HδA and if x ∈W then x is

not ‘stable below µ’;

5. By definition then, HδA ⊆ Hµ

A and x ∈W entails x 6∈ HµA.

Proof of the Stability Lemma: Details

Notice that if δ is an ordinal given by covering, we have that

x ∈ H+∞A ↔ ∀β(δ ≤ β → x ∈ Hβ

x ∈ H−∞A ↔ ∀β(δ ≤ β → x 6∈ HβA)

are satisfied.

This motivates the following (Σ1–)definition of unstable elements:

Definition

For every n ∈ N and δ arbitrary but fixed ordinal given by covering,we say that n is unstable (relatively to δ) (abbreviated: Uδ(n)) if

Uδ(n) := ∃β(δ ≤ β ∧ n ∈ HβA) ∧ ∃γ(δ ≤ γ ∧ n 6∈ Hγ

Notice that if δ is an ordinal given by covering, we have that

x ∈ H+∞A ↔ ∀β(δ ≤ β → x ∈ Hβ

x ∈ H−∞A ↔ ∀β(δ ≤ β → x 6∈ HβA)

are satisfied.This motivates the following (Σ1–)definition of unstable elements:

Definition

For every n ∈ N and δ arbitrary but fixed ordinal given by covering,we say that n is unstable (relatively to δ) (abbreviated: Uδ(n)) if

Uδ(n) := ∃β(δ ≤ β ∧ n ∈ HβA) ∧ ∃γ(δ ≤ γ ∧ n 6∈ Hγ

We first need a function enumerating W = n ∈ N | Uδ(n):

F (0) := minN z ∈ N.Uδ(z)

F (α) :=

minN z ∈ N.Uδ(z) ∧ (∀β < α)(F (β) < z), if it existsF (0), otherwise

F does this w.r.t. the given order of N.

We have:

I W is a Σ1 set, the totality of F simply follows by theΣ2–recursion theorem.

I F exhausts W in ω steps, and it keeps exhausting it afterevery ξ number of steps afterwards, for ξ limit ordinal.Elements of W occurr infinitely often in a list provided by F ifwe set dom(F ) = γ, where γ is such thatLim+(γ) := 0 < γ ∧ (∀α < γ)(∃β < γ)(α < β ∧ Lim(β))

F (α) :=

F does this w.r.t. the given order of N.We have:

F (α) :=

F does this w.r.t. the given order of N.We have:

Let then λ be an ordinal such that Lim+(λ) holds. We define:

G (0) = δ

G (α + 1) =

minµ.G (α) < µ ∧ F (α) ∈ Hµ

A, if F (α) 6∈ HG(α)A

minµ.G (α) < µ ∧ F (α) 6∈ HµA, otherwise

G (ξ) = minµ. sup(G (β) | β < ξ) < µ, ξ limit

and G is a provably total ∆T2 –function (as F is).

Forµ0 = min ξ.∀γ((∃β < λ)(γ = G (β))→ γ < ξ)

one has:

I µ0 is a limit ordinal satisfying the property given by thecovering lemma;

I G is strictly increasing below λ (hence, below m0);I members of W behave as unstable elements, hence they are

not retained at Hµ0

A ,H−µ0

A . Q.E.D.

G (0) = δ

G (α + 1) =

one has:

I µ0 is a limit ordinal satisfying the property given by thecovering lemma;

A ,H−µ0

A . Q.E.D.

G (0) = δ

G (α + 1) =

one has:I µ0 is a limit ordinal satisfying the property given by the

covering lemma;

A ,H−µ0

A . Q.E.D.

G (0) = δ

G (α + 1) =

covering lemma;I G is strictly increasing below λ (hence, below m0);

I members of W behave as unstable elements, hence they arenot retained at Hµ0

A ,H−µ0

A . Q.E.D.

G (0) = δ

G (α + 1) =

covering lemma;I G is strictly increasing below λ (hence, below m0);I members of W behave as unstable elements, hence they are

A ,H−µ0

A . Q.E.D.

On theories for quasi-inductive definitions

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